On the multiplication of free N-tuples of noncommutative random variables

Let a₁,..., a_n, b₁,..., b_n be random variables in a noncommutative probability space, such that {a₁,..., a_n} is free from {b₁,..., b_n}. We show how the joint distribution of the n-tuple (a₁b₁,..., a_nb_n) can be described in terms of the joint distributions of (a₁,..., a_n) and (b₁,..., b_n), by using the combinatorics of the n-dimensional R-transform. We point out a few applications that can be easily derived from our result, concerning the left-and-right translation with a semicircular element (see Sections 1.6-1.10) and the compression with a projection (see Sections 1.11-1.14) of an n-tuple of noncommutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper.